The implementation of a Generalized Cross Validation algorithm using deflation techniques for linear systems

Abstract

The fitting of a thin plate smoothing spline to noisy data using the method of minimizing the Generalized Cross Validation (GCV) function is computationally intensive involving the repeated solution of sets of linear systems of equations as part of a minimization routine. In the case of a data set of more than a few hundred points, implementation on a workstation can become unrealistic and it is then desirable to exploit high performance computing. The usual implementation of the GCV algorithm performs Householder reductions to tridiagonalize the influence matrix and then solves a sequence of tridiagonal linear systems which are updated only by a scalar value (the minimization parameter) on the diagonal. However, this approach is not readily parallelizable. In this paper the deflation techniques described in Burrage et al. (1994), which are used to accelerate the convergence of iterative schemes applied to linear systems, will be adapted to the problem of minimizing the GCV function. This approach will allow vector and parallel architectures to be exploited in an efficient manner.